Temperature compensated sapphire resonator for ultrastable oscillator operating at temperatures near 77° Kelvin

ABSTRACT

A sapphire resonator for an ultrastable oscillator capable of substantial performance improvements over the best available crystal quartz oscillators in a compact cryogenic package is based on a compensation mechanism enabled by the difference between copper and sapphire thermal expansion coefficients for so tuning the resonator as to cancel the temperature variation of the sapphire&#39;s dielectric constant. The sapphire resonator consists of a sapphire ring separated into two parts with webs on the outer end of each to form two re-entrant parts which are separated by a copper post. The re-entrant parts are bonded to the post by indium solder for good thermal conductivity between parts of that subassembly which is supported on the base plate of a closed copper cylinder (rf shielding casing) by a thin stainless steel cylinder. A unit for temperature control is placed in the stainless steel cylinder and is connected to the subassembly of re-entrant parts and copper post by a layer of indium for good thermal conduction. In normal use, the rf shielding casing is placed in a vacuum tank which is in turn placed in a thermos flask of liquid nitrogen. The temperature regulator is controlled from outside the thermos flask to a temperature in a range of about 40° to 150° K, such as 87° K for the WGH 811  mode of resonance in response to microwave energy inserted into the rf shielding casing through a port from an outside source.

ORIGIN OF THE INVENTION

The invention described herein was made in the performance of work undera NASA contract, and is subject to the provisions of Public Law 96-517(35 U.S.C. 202) in which the Contractor has elected not to retain title.

TECHNICAL FIELD

This invention relates to a type of sapphire resonator commonly called a"Whispering Gallery" sapphire resonator (hereinafter referred to as a WGsapphire resonator or simply a sapphire resonator) designed for thedominant (WGH_(n11)) microwave mode family to be temperature compensatedfor frequency stable operation at temperatures near 77° K (i.e., in arange of about 40° to 150° K).

BACKGROUND ART

A WG sapphire resonator consists of a ring or disk of sapphire inside ametallic cylindrical casing for electromagnetic shielding of andconfining resonating rf fields to the sapphire element. These resonatorseffectively eliminate rf conduction losses and thus make possibleoscillators that are only limited by performance of the sapphire itself.The sapphire is typically oriented with its crystal c-axis along theaxis of the cylindrical casing in order to achieve cylindrical symmetryfor the excited electromagnetic resonance modes.

WG electromagnetic modes can be divided into families depending on theirfield configuration, and further characterized by the number n of fullwaves around the perimeter of the sapphire ring or disk. The modes aredoubly degenerate, with azimuthal phase of the two submodes differing by90°. Modes typically used are the WGH_(n11) family for ring resonatorsand the WGE_(n11) family for flat disk resonators, where n≧5. WG denotesWhispering Gallery and H_(n11) denotes electric field loops formed inthe annular body of a wheel or ring, and E_(n11) denotes electric fieldloops formed in the planar body of a sapphire disk.

With very high microwave quality factors (Q's) at easily reachedcryogenic temperatures, the sapphire resonators already make possibleexcellent phase noise performance. In principle, the high Q values alsomake possible high frequency stability, but only if the resonator itselfwere stable. Temperature fluctuations in the sapphire cause unwantedfrequency fluctuations in the resonator. If these frequency variationscould be cancelled or compensated, high stability could be achieved.

Q of the WG sapphire resonator increases rapidly as the temperature iscooled, from approximately Q=300,000 at 300° K (room temperature) to 30million at 77° K (for X-band frequencies≈8 GHz). This compares to Qvalues of 1 to 2 million for the best available crystal quartzoscillators, and 10,000 to 20,000 for metallic microwave cavities.Consequently, when coupled with low noise microwave circuitry, the highsapphire Q theoretically could make possible long term frequencystability as low as 10⁻¹⁴ were it not for unwanted temperaturefluctuations in the resonator casing. Such a stability would be 20 timesbetter than that achievable by quartz oscillators of the highestquality, which presently provide a stability of 2×10⁻¹³.

Various approaches for compensated operation have been developed toreduce thermal variations in electromagnetic or acoustic (piezoelectric)resonators in order to achieve high frequency stability. Compensatedoperation for bulk acoustic-wave quartz oscillators is achieved by meansof an appropriate choice of orientation for the quartz crystal. This ispossible due to a very strong variability of acoustic parameters withcrystal direction. Electromagnetic sapphire resonators have a muchsmaller anisotropy (≈35%) and no sign reversal for any of its thermaldependencies. In fact, up to the present time useful compensation ofsapphire resonators has only been possible at liquid heliumtemperatures, where incidental or added paramagnetic impurities give aneffective compensating effect. But helium temperature operation isexpensive, and impractical for most applications. A compensationmechanism for operation at 77° K or above would allow liquid nitrogen tobe used as the coolant in a very much smaller Dewar and less expensivecompensation mechanism.

Temperature sensitivity of the operating frequency is characteristic ofall electromagnetic and acoustic resonators due to thermally inducedvariation of the size, dielectric constants, speed of sound, etc., forsolid state materials. Fractional variation of these parameters istypically 10⁴ to 10⁵ parts per degree Kelvin. Consequently, achievingresonator stabilities of 10⁻¹³ to 10⁻¹⁴ would require nanodegree controlof temperature stability, an impossible task. Yet such a high degree ofstability is desired for use as a stable local oscillator for an atomicfrequency standard (atomic clock) of the type disclosed by John D.Prestage in U.S. patent application Ser. No. 08/246,041 titled ExtendedLinear Ion Trap Frequency Standard Apparatus, now U.S. Pat. No.5,420,549. The majority of such frequency standards required for variouscommercial, scientific and military applications are based on quartzcrystal oscillators. A sapphire resonator has the potential for greaterstability in many such applications.

Available techniques for higher stability and reduced thermal variationin resonator frequencies are:

Very low cryogenic temperatures (T<10° K) can be used to "freeze out"the thermally-induced variations, which vary as a function of T³ as thetemperature of components varies. This technique has been successfullyapplied to super-conducting, superconductor-on-sapphire, and WG sapphireresonators. However, the very low helium temperature required makes suchsystems large and expensive, and therefore impractical for mostapplications.

An inherently weak tuning mechanism may be used at the lowesttemperatures to provide complete cancellation. In this way paramagneticimpurities can compensate the thermal variation in sapphire resonatorsfor T≦6° K, but again, operation at such low liquid helium temperaturesis impractical for most applications.

The differing thermal coefficients for various properties of theresonator components can be played against each other in such a waythat, for some operating temperature, thermal frequency variations arecompensated or cancelled. Piezoelectric quartz resonators arecompensated in this way by an appropriate orientation of this stronglyanisotropic crystal (e.g. "SC" or "AT" cut quartz resonators).Unfortunately, an orientation dependent cancellation does not occur forelectromagnetic resonators where the anisotropy is much smaller (i.e.,where the temperature dependencies vary by only about ≈30% as theorientation is changed).

A resonator may be constructed using several similar materials withcompensating thermal characteristics. For example, dielectric resonatorsfor oscillators are typically stabilized by use of several materialswith thermal dielectric variations of opposite sign.

A mechanical tuning mechanism may be driven by thermal expansioncoefficients of the construction materials. This mechanism has beenpreviously applied to a sapphire resonator at room temperature using ahighly re-entrant geometry to achieve very low phase noise and astability of 4×10⁻⁸ over a period of ten seconds. S. L. Abramov, Ye. N.Ivanov and D. P. Tsarapkin, "A Low-Noise Self-Excited MicrowaveOscillator with a Thermally Compensated Disk Dielectric Resonator,"Radiotechnika, No. 11, 81-83 (1988), reprinted in English, Telecom &Radio Engineering, Vol. 43, No. 12, pp. 127-129 (1990) and D. P.Tsarapkin, "An Uncooled Microwave Oscillator with 1-Million EffectiveQ-Factor," Proc. 1993 IEEE International Frequency Control Symposium,pp. 779-783 (1993)!. Ultrahigh frequency stability better than 7.4×10⁻⁷per degree Kelvin was probably precluded by attempting to operate atroom temperature with a design using brass and Invar which are alloyshaving poor thermal conductivity and using two brass parts joined by asliding (threaded) joint which also has a poor thermal conductivity,thus giving rise to temperature gradients in the compensation mechanism.

Material Properties

Factors affecting sensitivity of the sapphire resonator's frequency totemperature are:

Variation of the dielectric constants ε with temperature is the greatestfactor. As shown in FIG. 1, the dielectric constants vary by 80 to 140parts per million (PPM) per degree Kelvin at room temperature 300° K, asshown by graphs A and B for variations parallel and perpendicular to theresonator axis. The resulting change in frequency f is just half thisvalue, or 40 to 70 PPM per degree Kelvin (since f∝1/√ε).

The expansion coefficients of sapphire impact the frequency directlygiving rise to a frequency change of 5 to 6 PPM per degree Kelvin.

Thermal expansion of a copper rf shielding casing is a small butsignificant factor. Because microwave energy density at the walls of thecasing is greatly reduced (typically 100 to 10,000 times, to enable ahigh sapphire Q), the frequency sensitivity to casing size is reduced bythis same factor. Thus, the 15 PPM per degree Kelvin copper expansionshown in FIG. 2 is reduced to 0.15 PPM per degree Kelvin or less.

Short term thermal stability of approximately 1μ° K can be attained atroom temperature. However, even this very low variability, when coupledwith a sapphire fractional frequency df/f sensitivity of ≈6×10⁻⁵ perdegree K at room temperature or ≈1.25×10⁻⁵ per degree K at 77° K, givesfractional frequency variations of 1-6×10⁻¹¹ per degree K. These valuesof instability are substantially worse than attainable with excellentquality quartz oscillators.

On the other hand, if this sensitivity could be compensated to firstorder, the remaining response is a second-order effect of ≈10⁻⁷ per(degree K)². If the temperature could be kept within only 0.01 degree Kof the compensation point, the remaining linear sensitivity would be≦1×10⁻⁹ per degree K. Coupling this sensitivity value with theachievable temperature stability of 1μ° K would result in a frequencystability of ≦10⁻¹⁵, a very attractive prospect.

Because there are no available internal compensation mechanisms thatwould give an effectively unchanging dielectric constant in the sapphireitself, it is necessary to rely on a physical part of the resonatorstructure to compensate for the thermal effect of a different part. Inparticular, if the thermal expansion profile of a second componentmaterial were sufficiently different from sapphire, and if a mechanismcould be found to use this difference to give a frequency variation ofopposite sign to that of the sapphire itself, a practical compensatedresonator could be constructed.

A consequence of this type of thermal compensation design is that thevarious parts must be at the same temperature, i.e., in excellentthermal contact with each other. For example, a temperature differentialof 1μ° K, between the parts would give rise to the same large fractionalvariation (1-6×10⁻¹¹ per degree K) variation in frequency that is to beobviated by the compensation mechanism. Fortunately, sapphire has one ofthe highest thermal conductivities for any solid material in the 40° to150° K temperature range. Thus, sapphire could be mated with some otherhigh thermal conductivity material in a composite resonator with a veryshort thermal time constant and overall high thermal conductivity toprovide a sapphire resonator structure with high immunity to frequencyvariations due to variations in internal temperature gradients.

Since sapphire's expansion coefficient is relatively small compared tomost materials, a natural choice is for a second material with a greaterthermal expansion coefficient. FIG. 2 shows a comparison betweensapphire and copper, which is a likely candidate by virtue of its highthermal conductivity. The difference between sapphire and copper values,shown in FIG. 2, can be used to tune a variable sapphire resonator. Thatthen is the basis for the compensation mechanism of the presentinvention. It is useful to compare this difference with the temperaturecoefficient of the sapphire dielectric constants shown in FIG. 1. Such acomparison indicates that the compensation task is much easier attemperatures closer to 77° K than 300° K since dielectric coefficientvariations are strongly reduced as the temperature decreases from 300° Kto 77° K, while the copper-sapphire expansion difference is not sostrongly reduced.

A comparison of the magnitudes of the two effects shows that a veryprecise and effective tuning mechanism is required to achievecompensation. However difficult that may seem, at temperatures near 77°K the task is not impossible. The difference in temperature expansion ofcopper and sapphire due to their respective expansion coefficients σ_(c)and σ_(s) evaluated at a temperature of 77° K is given by: ##EQU1##where x is the resonator height while the dielectric tuning effect (onehalf of the dielectric constant variation, averaging perpendicular andparallel components) is ##EQU2## where ω is the frequency of a mode of asapphire resonator. Combining these two equations, the required tuningsensitivity is given by ##EQU3## That equation shows that a differentialthermal expansion between copper and sapphire could be used tocompensate the dielectric constant variation in sapphire at temperaturesnear 77° K if a mechanism could be found that is able to tune abouttwice (actually a ratio of 13.5 to 7) as much as it moves on afractional basis, comparing Hertz per Hertz with centimeters percentimeter.

It is worth noting that compensation at near room temperature is muchmore difficult. A comparison of FIGS. 1 and 2 shows that increasing thetemperature from 77° to 300° K increases the required compensationsensitivity given in Eq. (1) by approximately four times so thatcompensation at near room temperature for a high degree of stability onthe order of 10⁻¹⁴ Hz per Hz would require mechanical tuning sensitivityfor the resonator to be increased by approximately four times. Perhapsthis could be accomplished by the use of a material with a greatercoefficient of expansion, such as zinc, and/or by the use of relativelyextreme geometries. However, successful compensation at room temperaturehas not yet been reported by anyone skilled in the art. The objective ofthis invention is to achieve such a high order of stability at nearliquid nitrogen temperature (such as 87° K) with a mode Q of 10⁷.

SUMMARY OF THE INVENTION

A sapphire resonator cooled, for example, by liquid nitrogen (LN₂) or aninexpensive closed cycle cryocooler for operation at a chosentemperature in a range of about 40° to 150° K for use as an ultrastableoscillator includes a sapphire ring in an rf shielding casing havinghigh thermal conductivity, so that while the casing is cooled and thenregulated at the chosen operating temperature, such as 87° K, by atemperature regulator, additional heat from the resonating rf fields inthe sapphire ring may be readily conducted out of the shielding casing.However, due to high thermal frequency sensitivity of the sapphire ringand the inability of the temperature regulator to maintain absoluteconstancy of the sapphire temperature, a nonsapphire tuning element isprovided to compensate for any thermal frequency variation comprisingseparation of the sapphire ring into two equal annular parts, each parthaving a central web on a side opposite the other annular part, thusforming two re-entrant parts of the sapphire ring and a metal postbetween the webs separating the two re-entrant parts with a lengthselected to create a small gap between the two re-entrant parts that isstrongly tuned to a WGH_(n11) mode.

The metal for the post separating the re-entrant parts of the sapphirering is selected to have a greater thermal expansion coefficient thanthat of sapphire, a very short thermal time constant and overall highthermal conductivity. An example of such a metal is copper. Metalalloys, such as brass (a copper alloy) and Invar (a steel alloy) used byAbramov et al., supra, have poor thermal conductivity compared tosingle-element metals such as pure copper or zinc at 77° to 87° K and soallow substantial thermal gradients across the resonator assembly.Because of differences in coefficients of thermal expansion between thetwo alloys, a thermal compensation mechanism can succeed only to theextent that the primarily temperature dependent expansion of thesapphire and of the compensating mechanism for the gap between the twore-entrant parts of the sapphire ring change equally but with oppositeeffect on the resonant frequency. If the compensation mechanism employsthermal connecting parts of different single-element metals or ofdifferent alloys, the compensation will succeed only to the extent thetemperature of these parts follows each other. Once the gap is set forthe selected operating temperature, it is expected that during operationthe sapphire temperature will vary, thus affecting the resonantfrequency, but as the sapphire temperature increases the metal postexpands to increase the gap between the two re-entrant parts, thuscreating a compensating (equal and opposite) effect in the sapphireresonant frequency, thereby stabilizing the sapphire resonant frequencywithin the rf shielding casing.

The two re-entrant parts of the sapphire ring and the metal spacing postbetween the webs must be thermally connected at an interface with highthermal conductivity and supported within the rf shielding casing by astructure of very low thermal conductivity. For the support of theassembly within the rf shielding casing, a thin metal of very lowcoefficient of thermal conductivity is employed, such as a thin-wallcylinder of stainless steel (alloy of iron and other metals with smallpercentages of carbon). To obviate any temperature gradient at theinterface between the metal post and the two re-entrant parts of thesapphire ring, a gold-plated annular area on the webs of the re-entrantparts are indium soldered to the metal post for thermal integrity. Astainless steel bolt through the web of the upper re-entrant part of thesapphire ring, the post and the web of the lower re-entrant part of thesapphire ring holds the assembly of dual re-entrant parts with a gapspacing post in very good thermal contact with the housing of theelectric temperature regulator supported on the base of the rf shieldingcasing with a thin-wall stainless steel cylinder.

The novel features that are considered characteristic of this inventionare set forth with particularity in the appended claims. The inventionwill best be understood from the following description when read inconnection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a graph of temperature coefficients of the dielectricconstants of sapphire crystal for the components parallel (graph A) andperpendicular (graph B) to the c-axis.

FIG. 2 shows a graph of thermal expansion coefficients for copper andsapphire.

FIG. 3 illustrates diagrammatically a cylindrical cross-section of a WGsapphire resonator cooled in a liquid helium Dewar (not shown).

FIG. 4 illustrates the electric field configuration of E-fieldelliptical loops having their major axis on the center line of asapphire ring resonator operating in the WGH_(n11) mode where n=6.

FIG. 5 illustrates diagrammatically a temperature compensated WGsapphire resonator cooled in a liquid nitrogen Dewar (not shown) inaccordance with the present invention.

FIG. 6 is a diagram of one E-field loop showing elements ofseries-capacitance model for a sapphire ring resonator having a vacuumgap centered on the center line of the ring as shown in FIG. 4.

FIG. 7 is an exploded isometric view of the temperature compensated WGsapphire resonator of FIG. 5 with a supporting copper base plate butwithout a copper casing supported by the base plate over the sapphireresonator.

FIG. 8 is a graph of gap sensitivity estimated by circuit analysis andfinite element calculation.

FIG. 9 is a graph of frequency dependence on gap spacing forelectromagnetic modes from various families for the compensated WGsapphire resonator using finite element calculation.

FIG. 10 is a graph of frequency difference between experimental resultsand finite element calculations for several mode families.

FIG. 11 illustrates calculated lines and experimental points oftemperature tuning rates at 77° K for three dominant (lowest frequency)mode families in a sapphire/copper compensated resonator with a gap of0.050 cm.

FIG. 12 is a graph of frequency dependence of temperature compensationfor the WGH₈₁₁ mode at 7.23 GHz showing a turnover temperature near 87°K.

FIG. 13 is a graph of measured frequency stability for a temperaturecompensated sapphire oscillator.

DETAILED DESCRIPTION OF THE INVENTION

Before describing a preferred embodiment of the present invention, ahypothetical design will first be described with reference to FIG. 3 ofa basic microwave sapphire resonator without any temperaturecompensation mechanism but cooled in a liquid helium Dewar for frequencystability. A sapphire wheel or ring 10 having an outside diameter of 5.0cm and height of 2.0 cm is provided with a web 11 for supporting theresonator on a copper post 12 which is in turn supported on a copperbase 13 by a stainless steel cylinder 14 having a thin wall for thermalisolation. The bottom end of the post 12 houses a component 15comprising a temperature control sensor and a heating element within thestainless steel cylinder 14. A copper cylinder 16 and lid 17, togetherwith the base 13, form an rf shielding casing that houses the sapphireresonator assembly secured to the base 13 by a stainless steel bolt 18.

The rf shielding casing housing the sapphire resonator assembly isplaced in a vacuum tank (not shown) which is in turn placed in a liquidhelium Dewar (not shown) for cooling the vacuum tank. The high thermalconductivity of the rf shielding casing will allow the dielectricresonator 10 to be cooled to very near the surrounding liquid heliumtemperature. The component 15 thermally isolated from the copper base 13maintains the resonator temperature approximately constant, for example˜6° K, under control of external circuits (not shown). If the resonatoris instead operated at a higher temperature, such as 77° K, thesensitivity of the sapphire dielectric constant to temperature increasesto such an extent that small temperature variations which are present inall systems will prevent high frequency stability from being obtained.

Sapphire is a crystalline form of Al₂ O₃. Its hexagonal crystalstructure gives it a preferred c-axis in which direction the crystalexhibits different properties from those of its other two axes. Withonly a moderately high dielectric constant (ε≈10), sapphire has not beenproven to be a resonator unto itself. However, when formed in the shapeof a wheel or ring and placed inside the cylindrical rf shieldingcasing, resonating rf fields excited by an external microwave sourcethrough a coupling port 19 are confined to the sapphire ring. FIG. 4illustrates the distribution of elliptical E-field loops that areconfined to the sapphire ring. Note that the major axis of theelliptical E-field loops are centered on the dashed center line of thesapphire ring while operating as a WG resonator in the WGH₆₁₁ mode. Fourregions describe the radial field configuration of the WG mode ofresonance: an inner evanescent region (inside an inner imaginarycylindrical boundary in space having a slightly greater radius than theinner wall of the sapphire ring); an outer evanescent region(surrounding the outer wall of the sapphire ring out to an outerimaginary cylindrical boundary in space); a standing wave region betweenthe inner and outer evanescent regions; and traveling wave regions whichare sectors of standing waves, i.e., pie-shaped regions where standingwave fronts are formed.

FIG. 5 illustrates a WG resonator for operation at near liquid nitrogentemperature, for example 87° K in one embodiment of the presentinvention described below. It uses thermal expansion as an addednonsapphire tuning element to compensate for thermal frequency variationof the sapphire of 0.5 cm in diameter. The resonator consists of asapphire ring separated into two annular re-entrant parts 21 and 22 ofequal height approximately equal to half the 2.3 cm height of thesapphire ring 21, 22. The two re-entrant parts are separated by a copperpost 23 between webs 24 and 25 which, together with the two re-entrantparts 21 and 22, form a re-entrant sapphire ring. The post 23 isprovided with a length that creates a small gap 26 of 0.05 cm betweenthe two re-entrant parts selected for the dominant WGH₈₁₁ microwave modein the example selected of operation at 87° K. Other modes at othertemperatures will require a different gap, which may be predeterminedbefore assembling the re-entrant parts on the spacing post 23.

If the 0.5 cm gap 26 between the two re-entrant parts is small theresonant frequencies of some of the WG modes are strongly tuned as thegap spacing changes. However, for available materials, a weak tuningeffect would result if the post were only as long as the gap spacing,such as if the webs 24 and 25 were both at the opposite ends of theirrespective re-entrant parts 21 and 22, i.e., such as if the twore-entrant parts 21 and 22 had been formed by cutting the ring 10 ofFIG. 3 through its centerline to create the gap 26 such that only a thindisk would be required for the post 23 to set a gap that separates theparts. For that reason each of the re-entrant parts 21 and 22 must bemade with respective webs 24 and 25 as shown so that the post 23 mayhave a length almost equal the height of the entire sapphire resonatorring with a gap between the two re-entrant parts. Thus, with the twore-entrant parts in effect forming a resonant ring, a strong thermaltuning effect is achieved due to the difference between the thermalexpansion coefficients of the copper post 23 and the sapphire re-entrantparts 21 and 22. The thermal expansion of the sapphire is a relativelyminor effect, except that it does subtract from the compensating motiongenerated by the copper post. What is being compensated by the greaterthermal expansion per degree Kelvin of the copper post, as shown in FIG.2, is the thermal variation of the dielectric constant of the sapphire.By properly adjusting the length of the post 23 to the height of theresonant ring formed by the parts 21 and 22 with the gap 26 betweenthem, the resonant tuning due to thermal expansion coefficients of thecopper post will completely cancel the sapphire's inherent variation ofthe sapphire dielectric constant due to changes in temperature byintroducing a variation in the gap through variation in the length ofthe post as the temperature changes. This complete cancellation of anyvariation in the sapphire dielectric constant due to temperaturevariations can be assured by calculating the post length required by theratio of the coefficients of thermal expansion for the copper post 23and the sapphire parts 21 and 22 and then adjusting the thickness of thewebs 24 and 25 required to establish the small gap of 0.05 cm.

The design frequency of the WG resonator is established by other designparameters, including the mode that is excited through a port 20. Thatis accomplished by those skilled in the art using conventional tools,such as finite element programmed computational techniques, which arenot a part of this invention that relates instead to architecture of themechanism needed for providing temperature compensation to the degree ofstability required of the resonant frequency for many applications. Ifthe post is made of copper, this architecture may be used forcompensation at temperatures near 77° K (in a range of about 40° to 150°K). For operating temperatures higher than, for example 87° K, selectedmaterials with higher expansion coefficients (e.g. zinc) could be usedfor the post 23. Similarly, for lower operating temperatures, materialswith lower expansion coefficients may be used.

As in the dielectric resonator of FIG. 3, the resonator subassembly ofFIG. 5 consisting of sapphire parts 21, 22 and post 23 is provided witha component 27 (temperature sensor and heating element) supported on acopper base plate 28 by a thin-wall stainless steel cylinder 29.Although the component 27 is shown in thermal contact with the post 23,it would be sufficient to heat that post through the web of the lowerre-entrant part 22. A copper cylinder 30 and copper top plate 31 form anrf shielding casing with the base plate 28 to complete the WG sapphireresonator assembly that then need only be placed in a vacuum tank, afterwhich the vacuum tank is placed in a liquid nitrogen Dewar for cooling.

While good thermal conduction must be provided between the post 23 andthe two sapphire re-entrant parts for good performance, it is importantto maintain good thermal isolation of the resonator assembly supportedon the base plate 28 for optimum performance. The stainless steelcylinder 29 used for support of the resonator assembly within theshielding casing has a very low coefficient of thermal conduction, butto further enhance the thermal isolation sought, further thermalisolation means may be provided such as by a dual-wall base withsignificant space between two walls of the base.

In order to achieve high frequency stability, the high Q of the sapphireresonator must not be degraded by the post 23 while good thermalconduction between the post and the sapphire re-entrant parts 21 and 22must be maintained in order to eliminate any temperature gradientbetween the post and the re-entrant parts. Good thermal conduction maybe assured by first providing a gold-plated annular area formed by vapordeposition, for example, on the surfaces of the re-entrant parts 21 and22 abutting the post 23. In addition a copper sleeve is press-fittedinto the axial aperture provided for a bushing 32 (optional) andstainless steel bolt 33 passing through the center of both re-entrantparts and the temperature regulation component 27 into a threaded holein the base plate 28. A corresponding reduced diameter portionprotruding from each side of the post 23 is press-fitted into thesleeves in the apertures of the re-entrant posts.

The bolt may alternatively pass through the regulation component 27 aswell and be threaded into the base plate 28 as shown or into a nut notin thermal contact with the base plate. The function of the bolt is tohold the sapphire resonator assembly in good thermal contact with thetemperature regulation component 27 which is controlled by externalmeans (not shown). FIG. 7 illustrates that bolt 33 in an explodedisometric view of a practical realization of the present inventiondescribed with reference to FIG. 5, where the reference numeralsintroduced for the various elements in FIG. 5 are retained for elementsin FIG. 7 without further description thereof. To complete the assemblyof the sapphire resonator comprising the two re-entrant parts 21, 22 andthe post 23, molten indium is applied to the abutting surfaces of there-entrant parts and pressure is applied to the re-entrant parts whilethe indium solidifies.

Thus, because the compensating tuning effects are due to the variablegap 26 (see FIG. 5) between the re-entrant parts 21 and 22, thermalgradients between those parts is minimized by providing high thermalconductivity from the temperature regulator through the post 23 to there-entrant parts. Consequently, good thermal conductivity is requiredbetween the regulation component 27 and the post 23. To assure that, alayer of indium (a soft metal) is provided on the abutting surface ofthe temperature regulator. As a cylindrical extension from that abuttingsurface is drawn into the sleeve in the aperture of the lower re-entrantpart 22 by the bolt 33, the soft layer of indium will flow in the jointbetween component 27 and the post 23 to assure good thermal conductivitythrough the joint. Alternatively, indium solder may be used to provide athermal connection to the lower re-entrant part 22 through its web 25.

It has been found that a copper post of approximately 20-30% of thesapphire ring diameter provides the required thermal conductivity, andan axial position for the post minimizes any degradation of theelectromagnetic energy in the selected WGH_(n11) mode concentrated nearthe outer perimeter of the sapphire ring formed by the re-entrant parts.

The compensation mechanism can be understood as follows. As thetemperature is raised, the mode frequencies tend to be lowered due tothe increasing dielectric constant and thermal expansion of the sapphirere-entrant parts reducing the gap. However, at the same time, the gap iswidened due to the large thermal expansion of the copper post. Theresulting increase in gap volume (dielectric constant ε=1 compared toε≈10 for sapphire) tends to raise the frequencies. These cancellingeffects can be balanced to provide complete compensation at sometemperature, which in this example has been chosen to be 87° K.

As previously discussed, the sapphire element itself is the primarytemperature-dependent element in this resonator. Thus, the compensatedcentral subassembly (consisting of the two sapphire re-entrant parts andthe copper post) is thermally isolated from the copper If shieldingcasing, and held at a stabilized operating temperature of 87° K byaction of a temperature regulator. The exact temperature selecteddepends on the experimentally determined turnover temperature for thesapphire resonator subassembly, e.g., 87° K as shown in FIG. 12. Thetemperature of the rf shielding casing must also be controlled foroptimal frequency stability, even though its thermal sensitivity is 100to 10,000 times reduced from that of the central subassembly. This isaccomplished by means of a second temperature regulating feedback system(not shown) that stabilizes the casing temperature to a value near 77 K.

Variation of Mode Frequency with Gap Spacing

As indicated in Eq. 1, the copper/sapphire composite resonator requiresa high tuning sensitivity in order to achieve compensation at atemperature near 77° K. The following presents two different approachesto evaluating the sensitivity of the frequency of the fundamentalWGH_(n11) mode to changes in a small gap at the resonator center plane.

With a knowledge of the mode configuration, first estimate thissensitivity using simple circuit models that incorporate the resonatordimensions in a natural way. This approach has the advantage ofilluminating the qualitative features of the design problem.

A programmed finite element computational technique can then be used toestimate mode frequencies, both with and without a gap. Accuracy of thismethodology depends on the number of nodes used to characterize thegeometry, with fields being evaluated only at the node points, and withthe space between fitted to a simple power law behavior.

It is known that the magnitude of the tuning can be relatively large, asrequired by Eq. 1, because electromagnetic boundary conditions can giverise to larger energy in the gap region than in the (high ε) sapphire.In particular, this is true for modes with large electric fieldsperpendicular to the gap such as the WGH_(n11) mode family chosen.

Gap Sensitivity Estimation by Circuit Analysis

In order to demonstrate that the above approach is sound, the circuitanalysis approach is used to estimate a lower bound for the tuningsensitivity. The WGH_(n11) mode is characterized by a chain ofapproximately elliptical loops of electric field in the z-φ plane (asshown in FIG. 4) linked by loops of magnetic field in the r-φ plane.Because of the continuity of the electric field lines and ofdisplacement current, the effect of the gap can be estimated using asimple series-capacitance model. Conventional circuit analysis gives theresonant frequency ω=√1/LC, where L and C are the inductance andcapacitance of the circuit, and assuming that any change in effectiveinductance is small: ##EQU4## where ω is the resonance frequency and Δωis the change of frequency as a function of ΔC, which is the change incapacitance C due to a change in temperature which in turn changes thedimension of the gap 26. Each electric field loop traverses a pathlength l through the respective re-entrant parts 21, 22 of the sapphirering with dielectric constant ε_(s) and then a gap distance d with ε=1as shown in FIG. 6. Because of the continuity of displacement current,the sapphire capacitance C_(s) can be approximated as C_(s) ∝ε_(s) /lwhere l is the loop length and the gap capacitance C_(g) as C_(g) ∝1/d.By combining these capacitances in series the dependence of thecapacitance C on the gap spacing d is estimated as: ##EQU5##

A lower bound for the tuning effect is estimated by assuming that theloops are as long as possible, touching both the resonator top andbottom. Approximating the elliptical loops as circular the loop lengthbecomes ##EQU6## where h is the sapphire resonator height. Combiningthese two equations in the limit of small d, shows the frequencysensitivity in terms of distance sensitivity to be ##EQU7## where d isthe dimension of the gap between the resonator top and bottom halves, or##EQU8## where δx≡δd/h is the fractional variation of the resonatorheight.

Since ε_(s) ≈10.5, a comparison of Eqs. 1 and 2 shows that the circuitmodel predicts a sensitivity more than sufficient to achieve completecompensation at 77° K. That is, the tuning sensitivity of ε_(s) /π≈3 islarger than the required value of 13.5/7 from Eq. 1.

Gap Sensitivity Estimation by Finite Element Calculation

FIG. 8 shows a comparison of the circuit model prediction withcalculations using a recently developed finite-element methodology. ThisCYRES 2-D programmed method takes advantage of sapphire's cylindricallysymmetric dielectric properties to allow a simplified and more accuratecalculation of WG mode frequencies and fields than was previouslypossible. The finite-element approach allows relatively complicatedgeometries to be easily treated, such as geometries shown in FIG. 7.

As expected, this more accurate calculation gives a larger tuning effectthan the (lower bound) circuit model prediction, but somewhatsurprisingly shows an additional effect. As shown in FIG. 8, the finiteelement method predicts that tuning effectiveness (the slope of thecurve) will be degraded for gaps as small as 0.02 of the resonatorheight.

This reducing sensitivity is shown to be due to a feature not includedin our simple circuit model. For larger gaps (and also for large n,where the wide elliptical loops for E as shown in FIG. 6 tend to benarrow and tall instead) finite element solutions exhibit a substantialhorizontal (azimuthal) electric field component near the gap, showingthat the gap capacitance C_(g) is bypassed by azimuthal displacementcurrents in the sapphire. That is, a more accurate circuit model wouldcontain an additional capacitance C_(sg) acting in parallel with C_(g).

The variation in tuning sensitivity with gap spacing provides a means toadjust its strength in order to match other requirements. Thus the gapmay be varied to provide compensated operation in a particular mode at aparticular temperature.

Not all mode families are found to be strongly affected by changes inthe gap spacing. This is to be expected, for example, for modes withvery small E fields in the gap region. FIG. 9 shows the frequencydependence on gap spacing for three examples of modes with differentcharacteristics for their electric fields, namely, partiallycompensated, uncompensated and compensated modes. In order of decreasingsensitivity to gap spacing they are:

The WGH₈₁₁ mode with a maximum of vertical E field in the gap shows arapid increase in frequency with increasing gap spacing.

The WGE₈₁₁ mode has a maximum of radial (horizontal) E field at the gapand shows a slight frequency increase.

The WGH₈₁₂ mode with a sign reversal of vertical E field at the center,and very small values in the gap shows almost no change in frequency.

These results quantitatively confirm that a strong vertical E field inthe gap region, as exhibited by the WGH_(n11) mode family, and displayedin FIG. 4, is essential to achieve high sensitivity to gap spacing. Forthis geometry it is also the "fundamental" mode family, showing thelowest microwave frequency and highest mode confinement for any givenazimuthal wave number n. This family is thus an ideal candidate for usein a composite compensated resonator. Similar effects have beenpreviously used to provide frequency variability for sapphire resonators(M. E. Tobar and D. G. Blair, "Analysis of a Low Noise TunableOscillator Based on a Tunable Sapphire Loaded Superconducting Cavity,"Proc. 45th Symposium on Freq. Control, pp. 495-499, (1991); and D. G.Santiago, G. J. Dick and A. Prata, Jr., "Mode Control of CryogenicWhispering-Gallery Mode Sapphire Dielectric-Ring Resonators," IEEETrans. MTT, Vol. 42, pp. 52-55, (January, 1994)).

Experimental Tests of Mode Frequencies

A resonator was constructed with configuration and dimensions as shownin FIG. 5, and with the parts mechanically and thermally bonded by meansof pure indium solder. A clean (scraped) molten indium pool on each endof the copper post was mated in turn to an evaporated gold layer on thesapphire parts. After cooling to 77° K, the frequency, Q, and couplingcoefficient was measured for each of 69 resonant modes from 6.6 GHz to10.75 GHz. This list was then preliminarily matched by frequency withthe finite element data. Analysis of the electromagnetic visualizationof the resonator cross section using the CYRES 2-D software conclusivelyidentified the experimental modes for each family.

FIG. 9 shows the excellent agreement between theory and experiment forthe three mode families previously discussed which are the WGH₈₁₂ familyfor the uncompensated mode, WGE₈₁₁ family for the partially compensatedmode which has a small effect but not enough to be useful, and WGH₈₁₁family for the compensated mode which has a significant compensationeffect in the range of zero to about 1% of gap spacing variation, i.e.,change in sapphire resonator height due to increase in gap spacing.. Thedata indicates a frequency difference of less than 0.4%.

Temperature Tuning Rates

A direct demonstration of the effectiveness of our compensationmechanism can be shown by experimental measurement of the rate offrequency change with temperature at 77° K. As shown in FIG. 10, theexperimental points with positive values of frequency change indicatemodes that are actually overcompensated at 77° K where the members ofthe mode families WGH_(n12), WGE_(n11) and WGH_(n11) are designated bythe numbers which range from n=7 to 10 for the last family listed, fromn=7 to 12 for the penultimate family listed and n=8 to 13 for the firstfamily listed. They will have turnover temperatures (completecompensation) above 77° K which is desirable since this allows forrelaxed cooling requirements. Negative values indicatesunder-compensation or even no compensation (if values are approximatelythe same as expected from the sapphire dielectric variation alone), anda zero value would indicate a turnover temperature at exactly 77° K.

A comparison of calculated and measured tuning rates in FIG. 10 showsexcellent agreement. Sensitivity to small changes in the gap spacing wascalculated with the finite element software. The results were combinedwith values for the expansion coefficients of copper and sapphire (FIG.2) and a fitted value for the sapphire dielectric temperaturedependence. As shown in FIG. 1, the dielectric variation values can beexpected to vary between 9.4 PPM/Kelvin (perpendicular) and 16.75PPM/Kelvin (parallel) at 77° K. However this represents data measured atkilohertz frequencies and may be modified at microwave frequencies.

The fitted values were 11 PPM/Kelvin and 10.5 PPM/Kelvin for theWGH_(n11) and WGH_(n12) mode families, respectively, and 7 PPM/Kelvinfor the WGE_(n11) family. It is to be expected that the WGE familieswould have a lower value, because the electric fields in this case arealmost entirely in the r-φ plane shown in FIG. 4, and so correspond tothe lower "perpendicular" results, while the values for the WGH modeshave a substantial fraction of their electric fields "parallel" to the zaxis.

Both calculated and experimental values in FIG. 11 show a weakening ofthe compensation effect for higher mode numbers n (higher frequencies)that was not predicted by the simple circuit model as discussed inrelation to FIG. 8. Thus, modes with n≧9 in the WGH_(n11) mode familyare under-compensated at 77° K, and so would require operation at alower temperature. However the n=8 modes at 7.23 GHz are just slightlyovercompensated, and so can be operated at a temperature near 77° K, asdesired. More detailed calculations show that higher frequencycompensated operation is possible using a smaller gap spacing where thecompensation effectiveness is not so strongly dependent on the modenumber.

Compensated Resonator Operation

One of the two WGH₈₁₁ modes was chosen for further study. This modeshowed the highest quality factor of any of the compensated modes withQ=1.8×10⁶. FIG. 12 shows a plot of the resonance frequency for this modeshowing a turn-over temperature of 87.09° K. A quadratic approximationin the vicinity of the peak gives: ##EQU9## where f is the resonantfrequency in Hertz.

A residual linear thermal coefficient due to imperfect temperatureadjustment δT=T-87.09 can be derived from the slope of the curve as##EQU10## Eq. (3) allows the thermal requirements that would allow sucha resonator to achieve its ultimate stability of δf/f≈10⁻¹⁴ to beestimated. If the temperature is held at the turnover temperature withan accuracy of δT=1 millidegree, the slope given by Eq. 3 will be lessthan 2.34×10⁻¹⁰ per Kelvin, requiring a stability of 43 micro-Kelvins toachieve 10⁻¹⁴ stability. Accuracy and stability are distinguished inthis discussion, because the δT accuracy needs to be held over arelatively long operational time period (possibly days or months), whilethe strength of the sapphire oscillators is in short-term stability.Thus, in order to achieve a stability of 10⁻¹⁴ for a time period of e.g.100 seconds, the temperature would need to be stable to 43 micro-Kelvinsfor 100 seconds, but could vary up to 1 milliKelvin over the time periodof operation. These requirements are easily met using conventionalthermal regulation technology as developed for use by other types offrequency standards.

Oscillator Stability

An oscillator was constructed, stabilized by the WGH₈₁₁ mode of thecompensated resonator. Preliminary oscillator tests were accomplishedwith open loop control of the resonator temperature, and with the casingtemperature not regulated, but determined by direct contact with aliquid nitrogen bath. The stability of the oscillator was characterizedusing a hydrogen maser frequency standard as reference.

As shown in FIG. 13, the Allan Deviation of frequency variation wasmeasured to be 1.2-3.5×10⁻¹³ for measuring times 1 second≦τ≦100 seconds.For the shorter measuring times (τ≦30 sec.), this stability is superiorto that of the best available quartz oscillator. There was a large butconstant frequency drift during the course of the measurements which maybe due to slow relaxation in the soft indium solder joints. A frequencystability 10⁻¹⁴ for this resonator is projected with stabilized cantemperature and with a mode quality factor of Q=10⁷.

Although particular embodiments of the invention have been described andillustrated herein, it is recognized that modifications may readilyoccur to those skilled in the art. For example, it is relatively easierto achieve compensation at lower temperatures, such as 60° or 70° K. Theresonator could be operated at a higher mode (n>8) or be made lessre-entrant using a shorter copper post to move the turnover temperaturefrom 87° K to a lower temperature, and vice-versa for operation at ahigher temperature. Other design parameters may be varied, such as thetemperature coefficient of the dielectric constant or the differencebetween the thermal coefficients of the post the dielectric ring byproper selection of materials. Consequently, it is intended that theclaims be interpreted to cover such modifications and equivalentsthereof.

What is claimed is:
 1. A temperature compensated dielectric resonatorcooled to an appropriate temperature selected in a range of about 40° Kto 150° K for use as an ultrastable oscillator operating in said range,comprisinga dielectric ring in a cylindrical rf shielding casing havinghigh thermal conductivity, said dielectric ring being separated into twoannular parts with a gap between said annular parts selected foroperation in a selected mode of a Whispering Gallery ring resonator ofits H_(n11) family of modes, at an appropriate temperature selected insaid range for said resonance mode, a metal post between said twoannular parts of said dielectric ring re-entrantly attached to saidannular parts so that the length of said metal post and the distancebetween the interface points between said post and said annular partsare substantially longer than said gap which separates said annularparts, the length of said metal post being selected for spacing saidannular parts with said gap, the metal of said post having a greaterthermal expansion coefficient than that of the material of saiddielectric ring, a very short thermal time constant and overall highthermal conductivity, means for thermally connecting annular parts andsaid metal post at an interface thereof with high thermal conductivity,means for supporting in said cylindrical rf shielding casing saidannular parts with said metal post thermally connected therebetween todefine a subassembly, said supporting means for said subassembly havinga very low conductivity to provide thermal isolation of the resonatorand the copper base plate of the casing, and means for temperaturecontrol thermally joined to said subassembly for maintaining saidsubassembly substantially at said appropriate temperature for saidselected WGH_(n11) resonating mode of said dielectric resonator.
 2. Atemperature compensated dielectric resonator as defined in claim 1wherein said material of said dielectric ring is sapphire.
 3. Atemperature compensated dielectric resonator as defined in claim 2wherein said appropriate temperature is near
 77. 4. A temperaturecompensated dielectric resonator as defined in claim 3 wherein saidappropriate temperature is 87 K and said selected Whispering Galleryresonating mode is H₈₁₁ at 7.23 GHz.
 5. A temperature compensateddielectric resonator as defined in claim 1 wherein said two annularparts comprising said dielectric ring each have a separate central websubstantially less in thickness than the height of said dielectric ring,said central web of each part being on a side opposite the other annularpart to define two re-entrant parts of said dielectric ring with a gapbetween said annular parts, said metal post being thermally attached tothe facing surfaces of said respective central webs of saidcorresponding annular parts, thereby providing said means for thermallyconnecting said annular parts and said metal post.
 6. A temperaturecompensated dielectric resonator as defined in claim 5 wherein saiddielectric ring has a mode quality factor of Q>10⁶ and the metal of saidpost is copper for high tuning sensitivity of said gap in order tostabilize the resonant frequency of said dielectric ring separated intotwo re-entrant parts, said metal post serving to vary said gap as thetemperature of said subassembly varies during operation, therebycreating a compensating effect in said resonant frequency in order toachieve stability of said sapphire resonator on the order of δf/f≈10⁻¹⁴.7. A temperature compensated dielectric resonator as defined in claim 6wherein said means for supporting said subassembly comprising thermallyconnected re-entrant parts and said metal post therebetween in saidcylindrical rf shielding casing includes a cylinder of metal having alow coefficient of thermal conductivity between said means fortemperature control and a base plate of said cylindrical rf shieldingcan, said means for temperature control being contained within saidcylinder of metal without any direct thermal connection to said baseplate, and said subassembly is supported over said means for temperaturecontrol with a high thermal conductivity layer of indium therebetweenfor assuring a thermal junction of high conductivity between saidtemperature regulator and said subassembly.